We are continuing to develop novel Magnetic Resonance (MR) based displacement imaging methods, such as Diffusion Tensor MRI (DT-MRI or DTI) as a means to probe tissue microstructure and to assess and diagnose neurological and developmental disorders in vivo. DT-MRI measures a diffusion tensor of water within tissue noninvasively. It consists of relating an effective diffusion tensor to the measured MR spin echo signal; estimating an effective diffusion tensor, D, in each pixel from a set of diffusion-weighted MR images; and calculating and displaying information derived from D. This information includes the local fiber-tract orientation, the mean-squared distance water molecules diffuse in any given direction, the orientationally-averaged mean diffusivity, and other scalar invariant quantities that are independent of the laboratory coordinate system. These scalar parameters are intrinsic properties of the tissue that we measure without requiring contrast agents or dyes. For example, one DT-MRI parameter, the orientationally-averaged diffusivity (or Trace), has been the most successful imaging parameter used to date to visualize an acute stroke in progress. Moreover, we have shown that DT-MRI is effective in identifying Wallerian degeneration often associated with chronic stroke. Studies with kittens have shown DT-MRI to be useful in following early developmental changes occurring in cortical gray and white matter, which are not detectable using other means. The development of a method to color-encode nerve fiber orientation in the brain by Sinisa Pajevic and Carlo Pierpaoli has allowed us to identify and differentiate anatomical white matter pathways that have similar structure and composition, but different spatial orientations. Color maps of the human brain clearly show the main association, projection, and commissural white matter pathways. They have also allowed detailed studies of the brain's structural anatomy to be performed, which was only possible previously using laborious, invasive histological methods. To assess anatomical connectivity between different functional regions in the brain, we also proposed and demonstrated a way to use DT-MRI data to trace out nerve fiber tract trajectories, which we called DT-MRI "tractography". This development was made possible by contributions by Sinisa Pajevic and Akram Aldroubi who implemented a general mathematical framework for obtaining a continuous, smooth approximation to the measured discrete, noisy, diffusion tensor field data. We have also developed non-parametric (bootstrap) methods for determining features of the statistical distribution of the diffusion tensor from experimental DT-MRI data. Another innovation has been the development of a "tensor variate" Gaussian distribution that fully describes the variability of the diffusion tensor in an idealized experiment, and can be used to improve the design and efficiency of DT-MRI experiments. These developments have allowed us to apply powerful hypothesis tests to address a wide variety of important biological and clinical questions that previously could only be tackled using ad hoc methods. We are currently addressing other key methodological issues that will enable us to perform quantitative longitudinal and multi-center DT-MRI studies. Recently, we have been developing more sophisticated mathematical models of water diffusion in tissues and have begun using these to infer additional microstructural information about tissue (primarily white matter in the brain) from MRI data. The composite hindered and restricted model of diffusion (CHARMED) framework is one example. Physical phantoms are also being developed to test and interrogate our mathematical models water diffusion in complex media.